Best proximity results: optimization by approximate solutions
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Abstract
In this paper we utilize a generalized weakly contractive mapping to establish some best proximity point results which are global optimization results for finding the minimum distances between two sets. Amongst many approaches to this problem, we adopt the approach where the problem is treated as that of finding global optimal approximate solution of the fixed point equation for the generalized weak contraction mapping. We use three control functions to define such mappings. The results are obtained in metric spaces with a partial ordering defined therein. There is a blending of analytic and order theoretic approaches in the proofs. The uniqueness is obtained by imposing some order theoretic conditions additionally. There are several corollaries. An illustration of the main theorem through an example is given which also shows that the corollaries are properly contained in the main theorem.
Keywords
partially ordered metric space best proximity point fixed point equation approximate solution optimizationMSC
47H10 54H10 54H251 Introduction and mathematical preliminaries
The program of the paper is to find best proximity pairs between two subsets of a metric space with a partial ordering. There are several works which utilize for that purpose nonself mappings in the following manner. Let A and B be two nonintersecting subsets of a metric space \((X, d)\). A mapping \(S \colon A \longrightarrow B\) realizes the best proximity pair \((x, Sx)\) if \(d(x, Sx) = d(A, B)\). In that case the point x is called a best proximity point of S and the problem of finding such a point is designated as best proximity point problem. This area of research has attracted attention in recent time which has resulted into the publication of a good number of papers as, for instances, those which are noted in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16].
The problem has two aspects. Primarily, it is a global minimization problem, where the quantity \(d(x, Sx)\) is minimized over \(x\in A\) subject to the condition that the minimum value is \(d(A, B)\). When this global minimum is attained at a point z, then we have a best proximity point for which \(d(z, Sz) = d(A, B)\). Another aspect is that it is an extension of the idea of fixed point to which it reduces in the cases where \(A \cap B\) is nonempty. This is the reason that fixed point methodologies are applicable to these category of problems. More elaborately, the problem can be treated as that of finding a global optimal approximate solution of the fixed point equation \(x = Sx\) even when the exact solution is nonexistent for \(A \cap B = \emptyset\), which is the case of interest here. We adopt the latter approach in this paper.
We use a generalized weak contraction in our results. Weak contraction was studied in partially ordered metric spaces by Harjani and Sadarangani [17]. In a recent result by Choudhury et al. [18], a generalization of the above result to a coincidence point theorem has been made using three control functions. More specifically, here we utilize a generalized weak contraction mapping defined with the help of three control functions for the purpose of obtaining the desired minimum distance. The above mentioned mapping is assumed to be defined from one set A to the other set B. Then under suitable conditions, by applying fixed point methodologies, we obtained a best proximity point of the above mentioned mapping which realizes the minimum distance. Several metric and order theoretic concepts are utilized in our results. The main result has four corollaries and an illustrative example. Separate order theoretic condition are imposed to ensure the uniqueness of the best proximity point in the main result. It is also shown that the corollaries are properly contained in the main theorem.
The following are the requisite mathematical concepts for the discussions in this paper.
Definition 1.1
(Pproperty [16])
In [1], Abkar and Gabeleh show that every nonempty, bounded, closed, and convex pair of subsets of a uniformly convex Banach spaces has the Pproperty. Some nontrivial examples of a nonempty pair of subsets which satisfies the Pproperty are given in [1].
Lemma 1.1
([10])
Let \((A, B)\) be a pair of nonempty closed subsets of a complete metric space \((X, d)\) such that \(A_{0}\) is nonempty and \((A, B)\) has the Pproperty. Then, \((A_{0}, B_{0})\) is a closed pair of subsets of X.
Definition 1.2
Definition 1.3
([4])
In case of selfmapping the above definition reduces to the definition of increasing mapping.
Definition 1.4
Definition 1.5
 (i)
if \(\{z_{n}\}\) is any nondecreasing sequence in X converging to z, then \(z_{n} \preceq z\) for any \(n \geq0\);
 (ii)
if \(\{z_{n}\}\) is any nonincreasing sequence in X converging to z, then \(z_{n} \succeq z\) for any \(n \geq0\).
2 Main results

\(\Gamma=\{\eta\colon[0, \infty) \longrightarrow[0, \infty), \eta \mbox{ is continuous and monotonic increasing}\}\);

\(\Lambda=\{\xi\colon[0, \infty) \longrightarrow[0, \infty), \xi \mbox{ is bounded on any bounded interval in } [0, \infty)\}\).
Now, we discuss some properties of some special type of functions in Λ.
Let \(\Theta= \{\theta\in\Lambda: \underline{\lim}\, \theta(z_{n}) > 0, \mbox{whenever } \{z_{n}\}\mbox{ is any sequence of nonnegative real}\mbox{ }\mbox{numbers converging to }l> 0\}\).
We note that Θ is nonempty. For an illustration, we define \(\theta_{1}\) on \([0,\infty)\) by \(\theta_{1}(x)=e^{2x}\), \(x\in[0, \infty)\). Then \(\theta_{1}\in\Theta\). Here we observe that \(\theta_{1}(0)=1 > 0\). On the other hand, if \(\theta_{2}(x)=x^{3}\), \(x\in[0, \infty)\), then \(\theta_{2}\in\Theta\) and \(\theta_{2}(0)= 0\).
Also, for any \(\theta\in\Theta\), it is clear that \(\theta(x)> 0\) for \(x>0\); and \(\theta(0)\) need not be equal to 0.
Let \(\Upsilon= \{\varphi\in\Lambda: \overline{\lim}\, \varphi (z_{n}) < l, \mbox{whenever } \{z_{n}\}\mbox{ is any sequence of nonnegative real}\mbox{ }\mbox{numbers converging to }l > 0\}\).
It follows from the definition that, for any \(\varphi\in\Upsilon\), \(\varphi(y) < y\) for all \(y > 0\).
Theorem 2.1
 (i)
for \(x, y\in[0, \infty)\), \(\eta(x)\leq\xi(y) \Longrightarrow x\leq y\),
 (ii)
\(\eta(z)  \overline{\lim}\, \xi(z_{n}) + \underline{\lim}\, \theta(z_{n}) > 0\), whenever \(\{z_{n}\}\) is any sequence of nonnegative real numbers converging to \(z> 0\),
 (iii)for all \(x, y, u, v \in A_{0}\)where \(M(x, y, u, v) = \max \{d(x, y), \frac{d(x, u) + d(y, v)}{2}, \frac{d(y, u) + d(x, v)}{2} \}\).$$\left . \textstyle\begin{array}{r@{}} x \preceq y, \\ d(u, Sx)= d(A, B),\\ d(v, Sy)= d(A, B) \end{array}\displaystyle \right \}\quad\Rightarrow\quad\eta \bigl(d(u, v)\bigr) \leq\xi\bigl(M(x, y, u, v)\bigr)  \theta \bigl(M(x, y, u, v) \bigr), $$
Suppose either S is continuous or X is regular. Also, suppose that there exist elements \(x_{0}, x_{1} \in A_{0}\) for which \(d(x_{1}, Sx_{0}) = d(A, B)\) and \(x_{0} \preceq x_{1}\). Then S has a best proximity point in \(A_{0}\).
Proof
It follows from the definition of \(A_{0}\) and \(B_{0}\) that for every \(x\in A_{0}\) there exists \(y\in B_{0}\) such that \(d(x, y) = d(A, B)\) and conversely, for every \(y'\in B_{0}\) there exists \(x'\in A_{0}\) such that \(d(x', y') = d(A, B)\). Since \(S(A_{0})\subseteq B_{0}\), for every \(x \in A_{0}\) there exists a \(y \in A_{0}\) such that \(d(y, Sx) = d(A, B)\).
Next we show that \(\{x_{n}\}\) is a Cauchy sequence.

Suppose that S is continuous.

Next we suppose that X is regular.
Theorem 2.2
In addition to the hypotheses of Theorem 2.1, suppose that for every \(x, y \in A_{0}\) there exists \(u \in A_{0}\) such that u is comparable to x and y. Then S has a unique best proximity point.
Proof
Let \(Q_{n} = d(u_{n}, x)\), for all \(n \geq0\).
With the help of Pproperty we have the following theorem which is obtained by an application of Theorem 2.1.
Theorem 2.3
 (i)
for \(x, y\in[0, \infty)\), \(\eta(x)\leq\xi(y) \Longrightarrow x\leq y\),
 (ii)
\(\eta(z)  \overline{\lim}\, \xi(z_{n}) + \underline{\lim}\, \theta(z_{n}) > 0\), whenever \(\{z_{n}\}\) is any sequence of nonnegative real numbers converging to \(z> 0\),
 (iii)for all \(x, y, u, v \in A_{0}\)where \(M(x, y, u, v) = \max \{d(x, y), \frac{d(x, u) + d(y, v)}{2}, \frac{d(y, u) + d(x, v)}{2} \}\).$$\left . \textstyle\begin{array}{r@{}} x \preceq y, \\ d(u, Sx)= d(A, B),\\ d(v, Sy)= d(A, B) \end{array}\displaystyle \right \}\quad\Rightarrow \quad\eta \bigl(d(Sx, Sy)\bigr) \leq\xi\bigl(M(x, y, u, v)\bigr)  \theta \bigl(M(x, y, u, v)\bigr), $$
Suppose either S is continuous or X is regular. Also, suppose that there exist elements \(x_{0}, x_{1} \in A_{0}\) for which \(d(x_{1}, Sx_{0}) = d(A, B)\) and \(x_{0} \preceq x_{1}\). Then S has a best proximity point in \(A_{0}\).
Proof
By Lemma 1.1, \(A_{0}\) is nonempty and closed. Since \((A, B)\) satisfies the Pproperty, \(d(u, Sx)= d(A, B)\) and \(d(v, Sy)= d(A, B)\) imply that \(d(u, v)= d(Sx, Sy)\). Then condition (iii) of the theorem reduces to the condition (iii) of Theorem 2.1. Therefore, all the conditions of the Theorem 2.1 are satisfied and hence we have the required proof. □
3 Corollaries and example
Corollary 3.1
Proof
Let η be the identity mapping and \(\theta (t) = 0\) for all \(t \in[0, \infty)\) in Theorem 2.1. Then we have the required proof from that of Theorem 2.1. □
Corollary 3.2
Proof
The required proof is obtained by considering ξ to be identical with the function η in Theorem 2.1. □
Corollary 3.3
Proof
Let η and ξ be the identity mappings in Theorem 2.1. Then we have the required proof from that of Theorem 2.1. □
Corollary 3.4
Proof
We consider that η and ξ are the identity mappings and \(\theta(t)=(1k)t\), where \(0\leq k < 1\), in Theorem 2.1. Then we have the required proof from that of Theorem 2.1. □
Example 3.1
The function S now satisfies all the postulates of Theorems 2.1 and 2.2. Then, by joint applications of Theorems 2.1 and 2.2 we conclude that S must have a best proximity point which is unique. The point can be seen here to be \((0, 1) \in A_{0}\).
Note
In this example A and B are not closed sets. This is an illustration of the fact that the closedness of A and B are not required in our theorem.
4 Conclusions
The present paper is an application of weak inequalities satisfied by nonself mappings. Weak contractions are intermediate to the contractions and nonexpansions which have been generalized in various ways and have been utilized in different types of problem. We make such an application for finding a best proximity pair. The speciality of this paper is that it has been obtained in the most general settings of a metric space without any special assumptions on this space.
Notes
Acknowledgements
The authors gratefully acknowledge the suggestions made by the learned referee.
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